A rounding error analysis of the joint bidiagonalization process with applications to the GSVD computation

The joint bidiagonalization(JBD) process is a useful algorithm for approximating some extreme generalized singular values and vectors of a large sparse or structured matrix pair {A,L\}. We present a rounding error analysis of the JBD process, which establishes connections between the JBD process and the two joint Lanczos bidiagonalizations. We investigate the loss of orthogonality of the computed Lanczos vectors. Based on the results of rounding error analysis, we investigate the convergence and accuracy of the approximate generalized singular values and vectors of {A,L\}. The results show that semiorthogonality of the Lanczos vectors is enough to guarantee the accuracy and convergence of the approximate generalized singular values, which is a guidance for designing an efficient semiorthogonalization strategy for the JBD process. We also investigate the residual norm appeared in the computation of the generalized singular value decomposition (GSVD), and show that its upper bound can be used as a stopping criterion.